Learn Spanish (Latin America): Avanzamos hacia la parada.

Yes, but it changes the meaning. hacia means you are moving in the direction of the stop ("towards"). If you change it to hasta la parada, you are setting a limit, meaning you advanced "as far as" or "up to" the stop and no further.

Is avanzamos in the present or past tense here?

It could be either! For regular -ar verbs like avanzar, the nosotros (we) form is exactly the same in the present tense ("we advance") and the preterite past tense ("we advanced"). You would need more context in a real conversation to know for sure.

Does la parada refer to a stop sign?

No, la parada is used for a public transit stop, like a bus stop (la parada de autobús). If you are talking about a physical stop sign on the street, it is usually called una señal de alto or simply un pare, depending on the country in Latin America.

Why don't we need a word for "we" before avanzamos?

In Spanish, the ending of the verb tells you who is doing the action. The -amos ending on avanzamos specifically means "we." You can add the word nosotros before it for emphasis, but it is completely optional and usually left out.